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<p>前面已经从哥伦布编码引出了指数哥伦布编码，对哥伦布编码有了一定的了解，在H264的SPS帧中就使用到了指数哥伦布编码，这里来详细的学习一下。</p>
<h1 id="一、指数哥伦布编码简介"><a href="#一、指数哥伦布编码简介" class="headerlink" title="一、指数哥伦布编码简介"></a><font size=3>一、指数哥伦布编码简介</font></h1><h2 id="1-指数哥伦布编码"><a href="#1-指数哥伦布编码" class="headerlink" title="1. 指数哥伦布编码"></a><font size=3>1. 指数哥伦布编码</font></h2><p>指数哥伦布（Exp-Golomb）编码是一种在音视频编码标准中经常采用的可变长编码方法，它是使用一定规则构造码字的变长编码模式。它将所有数字分为等大小不同的组，符号值较小的组分配的码长较短，同一组内符号长基本相等，并且组的大小呈指数增长。</p>
<p>指数哥伦布码的比特串分为“前缀”（prefix）和“后缀”（suffix）两个部分。它的逻辑结构为：</p>
<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">[N个0][1][INFO] 或者 [Prefix 前缀][1][Suffix 后缀]</span><br></pre></td></tr></table></figure>

<ul>
<li><p>prefix：对应连续的N个0；</p>
</li>
<li><p>surfix：为表示实际数值的信息位，其长度与 prefix 一样。</p>
</li>
</ul>
<p>编码后码长为2N + 1 + k，N为前缀长度，1为中间的1长度，N+k为后缀长度。</p>
<h2 id="2-H264中的指数哥伦布编码"><a href="#2-H264中的指数哥伦布编码" class="headerlink" title="2. H264中的指数哥伦布编码"></a><font size=3>2. H264中的指数哥伦布编码</font></h2><p>在H264码流中就会有指数哥伦布编码的存在，例如SPS帧：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241213164505373.png" alt="image-20241213164505373"  />

<p>可以看到里面就会出现ue(v)、se(v)等，其实还会有te(v)、me(v)，他们都是指数哥伦布编码：</p>
<table>
<thead>
<tr>
<th align="center">类型</th>
<th>说明</th>
</tr>
</thead>
<tbody><tr>
<td align="center">ue(v)</td>
<td>无符号指数哥伦布编码</td>
</tr>
<tr>
<td align="center">se(v)</td>
<td>有符号指数哥伦布编码</td>
</tr>
<tr>
<td align="center">te(v)</td>
<td>截断指数哥伦布编码</td>
</tr>
<tr>
<td align="center">me(v)</td>
<td>映射指数哥伦布编码</td>
</tr>
</tbody></table>
<p>其中ue(v)是其他变型算法的基础，其他算法的结果由ue(v)的结果进一步处理得到。关于指数哥伦布编码的相关内容，在H264的相关文档中也有说明：<a target="_blank" rel="noopener" href="https://gitee.com/sumumm/audio-and-video/raw/master/%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99/T-REC-H.264-202408-I-E.pdf">ITU-T Rec. H.264 (08&#x2F;2024) Advanced video coding for generic audiovisual services</a>中的9.1 Parsing process for Exp-Golomb codes。</p>
<h1 id="二、0阶指数哥伦布编码"><a href="#二、0阶指数哥伦布编码" class="headerlink" title="二、0阶指数哥伦布编码"></a><font size=3>二、0阶指数哥伦布编码</font></h1><p>指数哥伦布编码是一种较简单的编码方法，正常来说，它可以拓展至K阶，也即K阶指数哥伦布编码。而在H.264中使用的，是0阶指数哥伦布编码，也即K等于0。下面我们来学习一下0阶指数哥伦布编码。</p>
<h2 id="1-ue-v"><a href="#1-ue-v" class="headerlink" title="1. ue(v)"></a><font size=3>1. ue(v)</font></h2><h3 id="1-1-ue-v-编码过程"><a href="#1-1-ue-v-编码过程" class="headerlink" title="1.1 ue(v)编码过程"></a><font size=3>1.1 ue(v)编码过程</font></h3><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216110711312.png" alt="image-20241216110711312"  />

<p>以待编码码号code_num &#x3D; 3为例：</p>
<ul>
<li><p>第一步：将code_num +1, 即3+1 &#x3D; 4</p>
</li>
<li><p>第二步：将4写为二进制的形式：100</p>
</li>
<li><p>第三步：计算100的比特个数为3，在100前面写（3-1）个0，得到编码码字：00100</p>
</li>
</ul>
<p>会发现里面有一些连续的码字是一样长的，可以总结如下：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216111148209.png" alt="image-20241216111148209"  />

<h3 id="1-2-ue-v-解码过程"><a href="#1-2-ue-v-解码过程" class="headerlink" title="1.2 ue(v)解码过程"></a><font size=3>1.2 ue(v)解码过程</font></h3><p>那怎么解码？其实可以直接在<a target="_blank" rel="noopener" href="https://gitee.com/sumumm/audio-and-video/raw/master/%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99/T-REC-H.264-202408-I-E.pdf">ITU-T Rec. H.264 (08&#x2F;2024) Advanced video coding for generic audiovisual services</a>文档的9.1 Parsing process for Exp-Golomb codes这里找到：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216111959120.png" alt="image-20241216111959120"  />

<p>当然，观察编码过程其实也能看出这个规律。其中leadingZeroBits为中间1前面，0的个数。所以在解码的时候，如果遇到描述为ue(v)，则可以先数0的个数，数到1为止，其中0的个数即为leadingZeroBits。而公式中的read_bits( leadingZeroBits )，则为从中间1开始，往后顺序数leadingZeroBits个比特位。利用这个公式，就可以计算出codeNum的值。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216114101208.png" alt="image-20241216114101208"  />

<h3 id="1-3-实例分析"><a href="#1-3-实例分析" class="headerlink" title="1.3 实例分析"></a><font size=3>1.3 实例分析</font></h3><p>例如下面有一个SPS帧：</p>
<p><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/d9cd173ae4fdd9532e68b79235c9687b.png" alt="img"></p>
<p>具体参数什么含义看前面的SPS帧的相关笔记，这里分析几个ue(v)的数据来深入了解一下这个编解码的过程。</p>
<h4 id="1-3-1-00-00-00-01-67"><a href="#1-3-1-00-00-00-01-67" class="headerlink" title="1.3.1 00 00 00 01 67"></a><font size=3>1.3.1 00 00 00 01 67</font></h4><figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">00 00 00 01 : start code</span><br><span class="line">67          : nalutype=7 sps</span><br></pre></td></tr></table></figure>

<h4 id="1-3-2-4D-00-29"><a href="#1-3-2-4D-00-29" class="headerlink" title="1.3.2 4D 00 29"></a><font size=3>1.3.2 4D 00 29</font></h4><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216130808284.png" alt="image-20241216130808284"  />

<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">4D          : profile_idc=0x4D(77=main profile)</span><br><span class="line">00 29       : level_idc=0x29(41就是4.1)</span><br></pre></td></tr></table></figure>

<h4 id="1-3-3-8D"><a href="#1-3-3-8D" class="headerlink" title="1.3.3 8D"></a><font size=3>1.3.3 8D</font></h4><p>其实到这里就是要解完才知道是到哪个字节了，因为这里开始就有一个ue(v)指数哥伦布编码的成员了：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216131110023.png" alt="image-20241216131110023"  />

<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">8D = 1000 1101</span><br></pre></td></tr></table></figure>

<p>根据指数哥伦布编码的原理，首个1bit是seq_parameter_set_id编码后的数据，前面0个0，所以这里不需要向后读取，这里就是2^0-1+0&#x3D;0，所以seq_parameter_set_id&#x3D;0。</p>
<p>继续解下一个成员的值，看这里其实seq_parameter_set_id后面的成员受到profile_idc的影响，上面profile_idc &#x3D; 77，不在这里所以这里的if不成立，直接跳到：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216134611596.png" alt="image-20241216134611596"  />

<p>根据SPS成员的定义，后面还是一个ue(v)的数据，这个时候就是解析<code>000 1101</code>，发现1前面有3个0，所以从第一个1后面读3位，就是<code>101</code>，也就是5，所以code_num就是2^3-1+5&#x3D;12。所以log2_max_frame_num_minus4 &#x3D; 12。</p>
<p>这样8D这个数据就解析完了。</p>
<h4 id="1-3-4-8D"><a href="#1-3-4-8D" class="headerlink" title="1.3.4 8D"></a><font size=3>1.3.4 8D</font></h4><p>接下来的数据还是8D，上面解析完后，接下来还是个ue(v)类型的：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216134819478.png" alt="image-20241216134819478"  />

<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">8D = 1000 1101</span><br></pre></td></tr></table></figure>

<p>所以第一个bit就是pic_order_cnt_type的值，这里pic_order_cnt_type&#x3D;0，后面继续解log2_max_pic_order_cnt_lsb_minus4的值，它也是一个ue(v)类型的，所以这里就是继续解析<code>000 1101</code>，前面计算过了，就是12，所以这里的值也是12，即log2_max_pic_order_cnt_lsb_minus4&#x3D;12.</p>
<h2 id="2-se-v"><a href="#2-se-v" class="headerlink" title="2. se(v)"></a><font size=3>2. se(v)</font></h2><p>se(v)也称<strong>有符号指数哥伦布编码</strong>，所以当描述字为se(v)时，它的输出有可能为负。而且当描述字为se(v)时，它的输入为ue(v)解码过程解析出来的codeNum。意思是什么呢？如果遇到se(v)，需要先调用ue(v)，得出codeNum的值。然后调用se(v)的解析过程，se(v)的输出即为语法元素的值。</p>
<h3 id="2-1-se-v-编码"><a href="#2-1-se-v-编码" class="headerlink" title="2.1 se(v)编码"></a><font size=3>2.1 se(v)编码</font></h3><p>这个se(v)编码相关的东西就没去研究了，根据解码的过程，应该是先把符号取消，然后按照u(ve)编码的方式进行编码。</p>
<h3 id="2-1-se-v-解码说明"><a href="#2-1-se-v-解码说明" class="headerlink" title="2.1 se(v)解码说明"></a><font size=3>2.1 se(v)解码说明</font></h3><p>主要还是关注解码的过程，在<a target="_blank" rel="noopener" href="https://gitee.com/sumumm/audio-and-video/raw/master/%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99/T-REC-H.264-202408-I-E.pdf">ITU-T Rec. H.264 (08&#x2F;2024) Advanced video coding for generic audiovisual services</a>的9.1.1 Mapping process for signed Exp-Golomb codes有说明：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/04%E9%9F%B3%E8%A7%86%E9%A2%91/LV05-%E8%A7%86%E9%A2%91%E7%BC%96%E8%A7%A3%E7%A0%81/LV05-02-H264-12-02-%E6%8C%87%E6%95%B0%E5%93%A5%E4%BC%A6%E5%B8%83%E7%BC%96%E7%A0%81/img/image-20241216140628231.png" alt="image-20241216140628231"  />

<p>如上表所示，表中第一列codeNum为输入，第二列为输出。计算公式则为：</p>
<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">语法元素值 = (−1)k+1 Ceil( k÷2 )</span><br></pre></td></tr></table></figure>

<p>其中Ceil为向上取整，k为codeNum的值，代入即可计算出语法元素的值。</p>
<blockquote>
<p>参考资料：</p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/CrystalShaw/article/details/129388266">h264编码概述八(哥伦布编码ue(v))-CSDN博客</a></p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/easyhao007/article/details/109896846">Exp-Golomb 指数哥伦布编码-CSDN博客</a></p>
<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/TaigaCon/p/3571651.html">指数哥伦布编码 - TaigaComplex - 博客园</a></p>
<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/DwyaneTalk/p/4035206.html">H.264学习笔记6——指数哥伦布编码 - DwyaneTalk - 博客园</a></p>
<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/wangguchangqing/p/6297792.html">Golomb及指数哥伦布编码原理介绍及实现 - Brook_icv - 博客园</a></p>
</blockquote>

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